9 The Laplace Transform
9,The Laplace Transform
9.1 The Laplace Transform
(1) Definition
dtetxsX st)()( )( jswh e r e
(2) Region of Convergence ( ROC )
ROC,Range of? for X(s) to converge
Representation,
A,Inequality
B,Region in S-plane
9 The Laplace Transform
Example for ROC
Re Re
S-planeS-plane
Im Im
-a -a
9 The Laplace Transform
(3) Relationship between Fourier and Laplace
transform
sjjs
tj
st
jXsXorsXjX
dtetxjX
dtetxsX








|)()(|)()(
)()(
)()(
Example 9.1 9.2 9.3 9.5
9 The Laplace Transform
9.2 The Region of Convergence for Laplace Transform
Property1,The ROC of X(s) consists of strips parallel
to j-axis in the s-plane.
Property2,For rational Laplace transform,the ROC
does not contain any poles.
Property3,If x(t) is of finite duration and is absolutely
integrable,then the ROC is the entire s-
plane
9 The Laplace Transform
Property4,If x(t) is right sided,and if the line Re{s}=?0
is in the ROC,then all values of s for which
Re{s}>?0 will also in the ROC.
9 The Laplace Transform
Property5,If x(t) is left sided,and if the line Re{s}=?0
is in the ROC,then all values of s for which
Re{s}<?0 will also in the ROC.
x(t)
T2 t
e-?0t
e-?1t
9 The Laplace Transform
Property6,If x(t) is two sided,and if the line Re{s}=?0
is in the ROC,then the ROC will consist of
a strip in the s-plane that includes the line
Re{s}=?0,
9 The Laplace Transform
S-plane
Re
Re Re
ImIm
Im
R?L?L
R
9 The Laplace Transform
Property7,If the Laplace transform X(s) of x(t) is
rational,then its ROC is bounded by
poles or extends to infinity,In addition,no
poles
of X(s) are contained in the ROC.
Property8,If the Laplace transform X(s) For rational
Laplace transform,the ROC
does not contain any poles.
Property3,If x(t) is of finite duration and is absolutely
integrable,then the ROC is the entire s-
plane
9 The Laplace Transform
Property7,If the Laplace transform X(s) of x(t) is
rational,then its ROC is bounded by
poles or extends to infinity,In addition,
no poles of X(s) are contained in the ROC.
Property8,If the Laplace transform X(s) of x(t) is
rational,then if x(t) is right sided,the ROC
is the region in the s-plane to the right of the
rightmost pole,If x(t) is left sided,the ROC
is the region in the s-plane to the left of the
leftmost pole.
Example 9.7 9.8
9 The Laplace Transform
Appendix Partial Fraction Expansion
Consider a fraction polynomial:
)(
)(
)(
)(
01
2
2
1
1
01
2
2
1
1
mnw h e r e
asasasas
bsbsbsbsb
sD
sN
sX
n
n
n
n
n
m
m
m
m
m
m


Discuss two cases of D(s)=0,for distinct root
and same root.
9 The Laplace Transform
(1) Distinct root:
)())((
)(
21
01
2
2
1
1
n
n
n
n
n
n
sss
asasasassD





n
i i
i
n
n
n
m
m
m
m
m
m
s
A
s
A
s
A
s
A
sss
bsbsbsbsb
sX
1
2
2
1
1
21
01
2
2
1
1
)())((
)(


thus
9 The Laplace Transform
Calculate A1,
Multiply two sides by (s-?1):
n
n
s
sA
s
sAAsXs


)()()()( 1
2
12
11?
1|)()( 11 ssXsA
Let s=?1,so
isi sXsAi |)()(
Generally
9 The Laplace Transform
(2) Same root:
)())(()(
)(
211
01
2
2
1
1
nrr
r
n
n
n
n
n
ssss
asasasassD



n
n
r
rr
rr
nrr
r
m
m
m
m
m
m
s
A
s
A
s
A
s
A
s
A
ssss
bsbsbsbsb
sX








1
1
1
1
1
1
12
1
11
211
01
2
2
1
1
)()()(
)())(()(
)(
thus
),,2,1(
|)()(
nrri
sXsAi isi


For first order poles:
9 The Laplace Transform
r
n
n
r
r
r
r
r
s
s
A
s
A
sAsAAsXs
)]([
)()()()(
1
1
1
1
11112111




Multiply two sides by (s-?1)r,
For r-order poles:
1|)()( 111 s
r sXsASo
1|)]'()[( 112 s
r sXsA
1|)]()[()!1(
1 1
11

s
rr
r sXsrA
9 The Laplace Transform
9.3 The Inverse Laplace Transform




















dejXtx
or
dejX
jXFetx
etxF
jsdteetxjX
dtetxsX
tj
tj
t
t
tjt
st
)(
1
)(
2
1
)(
)(
2
1
)]([)(
])([
)()()(
)()(
So

j
j
st dsesX
jtx
)(2
1)(
9 The Laplace Transform
The calculation for inverse Laplace transform:
(1) Integration of complex function by equation.
(2) Compute by Fraction expansion.
General form of X(s):


n
i i
i
n
n
s
A
s
A
s
A
s
AsX
1
2
2
1
1)(

Important transform pair:
p o l er i g h ttue
p o l el e f ttue
s t
t
i i
i
),(
),(1
Example 9.9 9.10 9.11